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(I've asked this question at Math StackExchange here but haven't gotten any response, so I decided to take a shot here as well.)

In his paper "Axioms of Symmetry: Throwing Darts at the Real Number Line," Freiling suggests two additional axioms that are formally analogous to his axiom of symmetry (which he proves to be equivalent to the negation of CH given ZFC):

  • A-null: For any $f$ from the reals to null sets of reals (relative to the Lebesgue measure), there are $x,y$ such that $x\not\in f(y)$ and $y\not\in f(x)$
  • A-meagre: For any $f$ from the reals to meagre sets of reals, there are $x,y$ such that $x\not\in f(y)$ and $y\not\in f(x)$.

Freiling says (p.194) he doesn't know whether A-null and A-meagre are jointly formally inconsistent (presumably given ZFC). Does anyone know if this question has already been answered? (I've made no progress on (dis)proving it myself...)

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  • $\begingroup$ You can show that the combination of A-null, A-meager and $2^{\aleph_0}=\aleph_2$ is inconsistent. It follows from Frelling's characterization of A-null under $2^{\aleph_0}=\aleph_2$ and its analogue form for A-null, and the ZFC-provable relationships in Cichon's diagram. $\endgroup$
    – Hanul Jeon
    Commented Mar 4, 2023 at 1:02
  • $\begingroup$ It might be worth examining models of ZFC where the additivity number of the null and meager ideals is $\aleph_2$ and the covering number of both ideals is $\aleph_3$. I am unsure which model would satisfy these conditions. $\endgroup$
    – Hanul Jeon
    Commented Mar 4, 2023 at 1:10

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